The dual \( (Tr^{\mathcal {A}_2}_t)^{*}: ((QP_t)_{n}^{GL_t})^{*}\rightarrow H^{t, t+n}(\mathcal {A}_2, \mathbb {Z}_2)\) 
of \(Tr^{\mathcal {A}_2}_t\)  is called the \(t\) -th cohomological transfer.
In [1]}, Singer proves that \( (Tr_t^{\mathcal {A}_2})^{*}\)  is an isomorphism for \(t =1,2.\)  Afterwards, based on Kameko's thesis [2]}, Boardman [3]} states that \( (Tr^{\mathcal {A}_2}_3)^{*}\)  is also an isomorphism. These works tell us that Singer's transfer map is highly non-trivial, and so, it serves as a reliable tool for depicting the cohomology of the Steenrod ring. In higher cohomological degrees, Singer [1]} gives some calculations to show that the fourth transfer is an isomorphism up to a range of internal degrees, but the fifth transfer is not an epimorphism, from which he makes the following prediction.
